On additive representation functions of finite sets, I (Variation)
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ON ADDITIVE REPRESENTATION FUNCTIONS OF FINITE SETS, I (VARIATION) ´ s Sa ´rko ¨ zy1 Andra Department of Algebra and Number Theory, E¨ otv¨ os Lor´ and University P´ azm´ any P´eter s´et´ any 1/C, H–1117 Budapest, Hungary E-mail: [email protected] (Received April 13, 2011; Accepted July 18, 2011) [Communicated by Attila Peth˝ o]
Abstract If m ∈ N, Zm is the additive group of the modulo m residue classes, A ⊂ Zm and n ∈ Zm , then let R(A, n) denote the number of solutions of a + a′ = n with a, a′ ∈ A. The variation V (A) = max |R(A, n + 1) − R(A, n)| n∈Zm
is estimated in terms of the number of a’s with a − 1 6∈ A, a ∈ A.
1. Introduction Let G be an additive semigroup, let A ⊂ G, and for g ∈ G let R(A, g) denote the additive representation function defined as the number of solutions of a + a′ = g with a, a′ ∈ A. The first important theorems on additive representation functions were proved by Erd˝ os and Fuchs [1]: they proved two theorems saying that if A is an infinite set of positive integers, then the function R(A, n) (n = 1, 2, . . .) cannot be nearly constant in a certain sense. Since that many papers have been written on additive representation functions including a series of 5 papers written by Erd˝os, S´ ark¨ ozy and T. S´ os [2], [3], [4], [5], [6]. In particular, in [4] we studied the following problem: what assumption is needed to guarantee that |R(A, n + 1) − R(A, n)| is not bounded? We proved
Mathematics subject classification number : 11B99. Key words and phrases: additive representation function, variation. 1
Research partially supported by Hungarian NFSR, grants no. K67676 and K72731.
0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
´ ¨ A. SARK OZY
Theorem A. Let s1 < s2 < · · · and t1 < t2 < · · · be positive integers satisfying s k < tk ≤ and put A = {a1 , a2 , . . .} =
sk+1 −1 2
(for k = 1, 2, . . .),
S+∞
k=1 {sk , sk+1 , . . . , tk }.
Then we have
|R(A, n + 1) − R(A, n)| ≤ 3 for all n. This theorem shows that if A consists of a “few” blocks of consecutive integers, then |R(A, n + 1) − R(A, n)| can be bounded (independently of the rate of growth of the counting function A(n) = |{a : a ∈ A, a ≤ n}| of A). On the other hand we proved that if the number of these blocks up to n, i.e., the function B(A, n) = |{a : a ≤ n, a − 1 ∈ / A, a ∈ A}| increases fast enough, then |R(A, n + 1) − R(A, n)| cannot be bounded: Theorem B. If A 6= ∅, then N X
(R(A, n + 1) − R(A, n))2 = o((B(A, N ))2 )
n=1
cannot hold. Corollary A. If A 6= ∅, then B(A, N ) max |R(A, n + 1) − R(A, n)| = o n≤N N 1/2 cannot hold. (Note that in [4] this equation appears with a typo on the right-hand side; the correct form is the one above.) Corollary B. If lim
N →+∞
B(A, N ) = +∞, N 1/2
then |R(A, n + 1) − R(A, n)| cannot be bounded. We also showed that Theorem B is nearly best possible:
ON ADDITIVE REPRESENTATION FUNCTIONS OF FINITE SETS
Theorem C. For all ε > 0 there exists an infinite sequence A such that 1
B(A, N ) ≫ N 2 −ε and R(A, n) is bounded (in fact
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